Advanced book on Mathematics Olympiad

Geometry and Trigonometry 651 A D B C E FP M N Figure 93 A(EP D)+A(N P DC)+A(BN C)=A(EN F )+A(AMF )+A(MN BA). AddingA(MN P )to b ...

652 Geometry and Trigonometry l− 0. 002 ≤PQ,PR,RP≤l+ 0. 002. On the other hand, since there is no equilateral triangle whose ver ...

Geometry and Trigonometry 653 658.The relation from the statement can be transformed into tan^2 b= tan^2 a+ 1 tan^2 a− 1 =− 1 co ...

654 Geometry and Trigonometry as desired. (Gazeta Matematica ̆(Mathematics Gazette, Bucharest), proposed by D. Andrica) 661.We w ...

Geometry and Trigonometry 655 cosx,cosy,cos(x+ky)can be obtained by adding polynomials of lower degrees, and eventually multiply ...

656 Geometry and Trigonometry Herex=sina,y=cosa,z=−^15. It follows that eitherx+y+z=0orx=y=z. The latter would imply sina=cosa=− ...

Geometry and Trigonometry 657 So we have to prove that ∏n k= 0 1 +bk 1 −bk ≥nn+^1. The inequality from the statement implies 1 + ...

658 Geometry and Trigonometry 668.Denote the sum in question byS 1 and let S 2 = ( n 1 ) sinx+ ( n 2 ) sin 2x+···+ ( n n ) sinnx ...

Geometry and Trigonometry 659 which means that cottjsint= 1 r −cost, forj= 1 , 2 , 3. If sint =0, then cott 1 =cott 2 =cott ...

660 Geometry and Trigonometry S^2 = 2 −^2 n ∏^2 n k= 1 (ζk+ζ−k)= 2 −^2 n× ∏^2 n k= 1 ζ−k× ∏^2 n k= 1 ( 1 +ζ^2 k). The first of t ...

Geometry and Trigonometry 661 674.By eventually changingφ(t)toφ(t)+θ 2 , whereθis the argument of 4P^2 − 2 Q, we may assume that ...

662 Geometry and Trigonometry so 2 sin 7 a 4 sin 5 a 4 = 0 , meaning thata = 0 ,^47 π,^45 π. It follows that the solutions to th ...

Geometry and Trigonometry 663 =sinusin(v+w)−cosucos(v+w)=−cos(u+v+w). And of course this takes values in the interval[− 1 , 1 ]. ...

664 Geometry and Trigonometry And this is Jensen’s inequality applied to the tangent function, which is convex on( 0 ,π 2 ). 682 ...

Geometry and Trigonometry 665 2 tanun+ 1 =bn+ 1 = 2 tanun 2 + √ 4 +4 tanun = 4 tanun 2 +cos^2 un = 2 · sinun 1 +cosun =2 tan un ...

666 Geometry and Trigonometry xn=tan ( 90 ◦− 30 ◦ 2 n−^1 ) =cot ( 30 ◦ 2 n−^1 ) =cotθn, whereθn= 30 ◦ 2 n−^1 . A similar calcula ...

Geometry and Trigonometry 667 n √ a+ √ a^2 − 1 + n √ a− √ a^2 − 1 =n √ cosht+sinht+n √ cosht−sinht =n √ et+n √ e−t=et/n+e−t/n=2 ...

668 Geometry and Trigonometry It follows that the left-hand side telescopes as 1 8 (3 tan 27◦−tan 9◦+9 tan 81◦−3 tan 27◦+27 tan ...

Geometry and Trigonometry 669 = lim N→∞ (arctan(N+ 1 )+arctanN−arctan 1−arctan 0) = π 2 + π 2 − π 4 = 3 π 4 . The sum in part (b ...

670 Geometry and Trigonometry nlim→∞Rn= ∏∞ n= 1 cos π 2 n . The product can be made to telescope if we use the double-angle form ...